# COVID-19 Event Risk Assessment Planner

The mathematics for calculating the probability of exposure given the number of carriers in a population and group size aren’t difficult but they can be surprising. Even a low number of carriers can generate a relatively high probability for reasonably sized groups. For example, assume you run a firm of 1000 people in the San Francisco Bay Area (population 8 million.) Let’s suppose that there are just 500 carriers in the area. In this case, assuming random draws, the probability that at least one of your employees is a carrier is 6%. You can run your own calculations at Wolfram Alpha following this format:

p=8000000, c=500, g=1000, 1-1(1-c/p)^g //N

where p is the population size, c is the number of carriers, g is the group size and the //N at the end isn’t a division but a command to Wolfram Alpha to give you a numerical answer.

Joshua Weitz on twitter put together this graphic using the same calculations but for the United States as a whole (population 330 million). It says, for example, that if there are 20,000 carriers in the United States then at a small concert of 1000 people there is a 5.9% chance of at least one carrier attending. At the March Madness final in Atlanta with 100,000 attendees there is a greater than 99% chance that at least one attendee is a carrier.

Now here is the most important point. It’s the size of the group, not the number of carriers that most drives the result. For example, suppose our estimate of the number of carriers if off by a factor of 10–that is instead of 20,000 there are just 2000 carriers in the United States. In this case, the probability of at least one carrier at a big event of 100,000 drops not by a factor of ten but just to 45%. In other words, large events are a bad idea even in scenarios with just a small number of carriers.

While we are working or going to school we have some base level of interactions, and associated fan out, with every worker and student and whatever they are getting up to.

If you are serious about social distancing, you don't want a dinner party with March Madness guy .. or a college professor.

Note that having dinner with your own family, or your own college professor, does not expand your interaction network.

Epidemics are basically revenge of the anti-socials/incels/loners/introverts. I'm waiting for the ev-psych crowd to defend the Darwinian selection for this bag of so-called "low value" traits. So-called "high social value" is a yuuge liability right about now.

The NEETs shall inherit the earth? Depressing take, but thankfully not accurate.

It’s closer to Boomers reaping the whirlwind/karma.

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It's time to invite your local hermit over for dinner.

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Ok at very large events, the probability that there’s at least one carrier present is high. However this carrier if not going to this large event ( it was canceled) may go to the pub instead . It’s not clear he/she will infect less people. China did not control the virus by banning very large gatherings but by pretty much banning all gatherings. I not advocating we do the same , just saying banning large events may do little.

Contact tracing can be somewhat easier if the interaction is in a smaller setting.

In Sacramento, the county health officer announced there would be no more quarantines and minimal tracing to determine how those testing positive for the novel coronavirus contracted the deadly, fast-moving disease. Those with symptoms face only voluntary restrictions for mingling in the community.

“The cat is already out of the bag,” said Peter Beilenson, Sacramento County’s health director. “The community spread has already occurred.”

That may mean that it's hit the homeless population, and they're worried about people stigmatizing them. Other places may have a different calculation still.

Has the National Guard been given orders to shoot anyone leaving New Rochelle, NY? If not, talk about how other places may have a different calculations.

prior, if you had any human contact offline, they could tell you when you're gibbering.

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Precisely. The contact-tracing phase is over. The testing phase is almost over. (Except for retrospective studies).

Its just isolation, cancellations, quarantines and treatment triage now.

The testing phase is definitely not over, as effective testing allows separating patients into two groups - those requiring more stringent isolation measures due to covid19, and those who don't.

This will become quickly relevant as PPE supplies dwindle - you do not want to be wasting such resources when treating people who are simply suffering from non-contagious lung problems.

Right, but unless the false negative problem is cured, testing doesn't really help, if it does free people who should be in more stringent isolation.

We obviously have a lot of problems right now, but I think one reason we don't test people in isolation is that they are in isolation.

Well, the testing is done as a first step, to screen patients requiring the more stringent measures.

The U.S. seems to be quite far from being able to do such initial routine screening at this point.

It is starting to look like we are on a third round of failures in terms of delivering testing at scale.

Meanwhile, a little quarantine humor:

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Re: The testing phase is almost over except for retrospective.

Agree. But, with the absence of test kits, retrospective analysis might be limited as well as to the number of undiagnosed carriers who become well later but nevertheless transmit.

So, why not have doctors take nose and throat swabs of their patients and preserve the swabs for later analysis as a way of getting a picture in time of what was the prevalence of covid in the population when tests were restricted? This might have benefits later for estimating n or p in the network contagion models.

This also may have utility if covid reinfects a person who previously had it--either leading to a overreaction by the immune system, or a collapse of it. This is data you might want to have.

You could even take your own nose and throat swab and put it in the freezer with the popsicles for testing at another time.

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Ok. I get that. I was referring more to capacity and timeliness. At some point I assume there isn't enough tests or enough time, and so they start assuming everyone has it. That's the mass-quarantine stage. We are in the mass-cancellation stage that comes before it. Treatment. meanwhile, is reduced to triage based on symptoms.

Yeah

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It’s not clear he/she will infect less people.

Actually it is. Unless the statistical liklihood of infection somehow rises at smaller events, then the expected number of infections should be proportional to the number of contacts.

The greatest risk of course is at event that least longer than a few days, because then one person can infect hundreds, who can then develop symptoms and infect hundreds more each.

For instance, the week-long potluck held in Wuhan in late January.

Or CPAC, where transmission may have already occurred, though it is not clear (for example, because the wife has yet to be tested, it seems, and the source of infection may have been different), as recounted in this Erik Wemple editorial from yesterday.

"Fran Finnegan, who lives in the San Francisco suburbs, was among the CPAC attendees quizzed by the Erik Wemple Blog about Mulvaney’s remarks. Would the media really do something this underhanded? Would it goose its coronavirus stories to hurt Trump’s chances at reelection? “Oh, absolutely. Part of the media. ... The liberal left media like your organization, New York Times, you know — that kind of stuff. The networks,” Finnegan said, adding that “to some extent” media bias affects the coverage of a potential pandemic.

Since returning home from CPAC, Finnegan has had cause to dig deeper into all things coronavirus. Last Tuesday, his wife started feeling ill, with symptoms including shortness of breath and coughing. This was extraordinary: In 32 years of marriage, says Finnegan, his wife had gotten sick once. “All of a sudden on Tuesday she started feeling terrible,” he recalls."

Yes, it's possible that someone could infect lots of people on Wednesday who then start becoming contagious on Saturday and infect hundreds more each. Big crowds of people standing close together for rallies - one sneeze exposes a couple of dozen people.

Keep in mind you're unlikely to catch it directly from another person except in conditions of intimate contact. Just saying Hi to someone on the street isn't likely to transmit the virus. The danger comes from touching hard surfaces where the virus has been deposited and then transferring it to nose, mouth or eyes. Every public space you enter exposes you to everyone who has been there for at least the last several hours. So the takeaway is Wash Your Hands, well and often
And limit surfaces you touch in public places (including your workplace).

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+1 You must know something about contagion models.

That +1 is meant for Hazel

Thanks, but no, I just know basic statistics.

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Pretty much banning all events is exactly what should be being advocated / done, at the very least in areas that have had significant outbreaks (e.g. Washington State). The actual number of cases are far higher than the officially confirmed cases, making contact-tracing & isolation an impractical undertaking (1,364 official US cases right now, so probably ~10k actual cases out there). Slowing the spread so that it can be dealt with effectively (i.e. health services are not overwhelmed) is the main option now, which means social distancing, i.e. no large events, stay home as much as possible, and wash your hands.

"With the number of cases in countries like the US, Spain, France, Iran, Germany, Japan or Switzerland, Wuhan was already in lockdown. And if you’re telling yourself: “Well, Hubei is just one region”, let me remind you that it has nearly 60 million people, bigger than Spain and about the size of France."

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Has anyone run the numbers yet for Liverpool's first championship in 30 years?

OK, did it myself... Assuming that only the population of Liverpool itself turns out to celebrate:

p=66440000, c=456, g=552267, 1-1(1-c/p)^g //N, gives a probability of 97.7%.

Need to take those numbers down a bit. There are a lot of Everton fans in Liverpool. Maybe look at how many celebrated in Edinburgh for Hibs' first Scottish Cup in 114 years.

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So how do the numbers stack together? Each flight from Europe is only a few hundred people, but the total over a month is thousands.

The number of Americans likely to be returning to America from Europe in the next several days will be quite large. And compared to a month ago, the number of carriers will also be higher. Both in the aircraft, and assuming those citizens are not placed into isolation for two weeks, in the U.S. itself.

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If you want to play with p (probability) or n (number of nodes), you might want to look at the "GEPHI" software programwhich lets you work with various network models, including contagion models. Here is the link for the free software program: https://gephi.org

There is a very good course on networks by Lada Adamic at U of Michigan available through Coursera.

A doctor friend and medical educator tells me that he limits the size of the groups he associates with to no more than ten people, and then only if necessary.

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"assume you run a firm of 1000 people...if there are 20,000 carriers in the United States then at a small concert of 1000 people there is a 5.9% chance of at least one carrier attending....It’s the size of the group, not the number of carriers that most drives the result."

That does not appear to be true at present US carrier numbers (~1000, not 20,000) and ~1000-person events. If one runs a 1000-person firm, then the probability that one of your employees is a carrier depends a lot on whether US carrier number is 2000 (0.6%) or 20,000 (6%).

Regardless of the probabilities, we can expect officials to be heavily biased towards cancelling events. If the subsequent outbreak is stopped/muted, then the cancellations will be credited for helping to slow/stop the growth of the disease. If the outbreak subsequently explodes, then people will be thankful that the cancellations reduced the number of people exposed to the disease. On the other hand, if events are not cancelled and the outbreak explodes, then officials will be blamed for not proactively cancelling events. So, it's easy to understand the wave of cancellations, closings, work-from-home orders, etc.

This chart, however, may help explain/guide behavior by private individuals, who are concerned with a priori risk/cost and benefits of attending non-cancelled events rather than avoiding ex-post blame. If an event is not cancelled, private individuals will indeed want to multiply these probabilities by conditional probability of catching disease given carrier present times cost of catching disease (??) and compare that against benefit of attending.

Uh..... I wouldn’t put so much faith in the official number of people infected since so few have been tested and there seem to be no plans to try to test a lot of people in areas with more cases as South Korea did. We’re flying blind. But don’t worry, we banned Europeans, except Britons of course since Coronavirus totally respects Brexit.

A fact that undoubtedly explains the current number of UK confirmed infections, which is 459.

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Is there a duty to disclose to attendees at Trump rallies that they have a 50% to 90% chance that there is one or more covid carriers in the audience.

Not funny, not enlightening.

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Though not precisely a Trump rally, CPAC has done an excellent job disclosing about a case of Covid19 and who might have been exposed.

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The market seems to have stabilized with the DJIA at 25,555, where at was yesterday evening.

Of course, this could change once the opening bell sounds.

i meant 23,555. No where to go but up.

Now is truly the time to buy - falling knife included free with every purchase.

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Yeah, so it's down another 2000 points to 21,500 as we speak.

About time that Trump removed another shameful reminder of the Obummer times.

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Buy now, and we will throw in a complete set of falling knives for free. After the circuit breakers reset, that is.

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Who knew that COVID would have such a devastating impact on the domestic American energy industry, which requires oil at around \$50 a barrel to more or less break even -

The US travel ban could pile further pressure on oil markets by cutting demand for jet fuel, according to Rystad Energy.

Oil prices fell to just above \$33 a barrel this morning after President Trump’s ban on European flights for the next 30 days.

The Norwegian energy analysts said traders received the news “very negatively” for three reasons: first, the cancelled flights could dent demand for oil by around 600,000 barrels; but importantly the move also stokes fear of further economic slowdown, and undermines confidence in the governments’ handling of the fall-out. The oil market was already reeling from a double-blow due to the impact of the coronavirus and plans for an oil price war between Saudi Arabia and Russia.

Rystad has warned that oil prices in the \$20s are not far off if the pair battle for dominance by pouring more oil into the market while demand remains weakened by Covid-19.

The fall off in oil demand will also make it more difficult for Russia and Saudi Arabia to come to an agreement on the allocation of now an even greater fall off in demand.

Everyone knows there are no winners in a price war other than the customers.

Will Iran directly or Russia indirectly start a crisis in the Middle East targeting Saudi oil supplies or shipping interests in order to raise oil prices.

Don't put it past them.

The oil industry most directly threatened is the domestic one.

A couple of smaples of what the Saudis have done to the U.S. supply side.

“Saudi Arabia is engaging in marketplace war fare, pure and simple, not missing an opportunity to make matters worse for its oil producing rivals, Russia and U.S. operators,” he said.

Another - The oil war, which may lower gasoline and fuel prices in the short term, could decimate the U. S.shale oil industry very quickly, putting millions out of work across oil, trucking and service industries.

Occidental Petroleum is one of the hardest hit U.S. oil companies; its stock fell 18 percent Wednesday. Oxy, as it is known, has seen its market value shrivel since its acquisition of Anadarko Petroleum last year. Its market cap is now below \$11 billion, compared with \$50 billion before the acquisition.

And something resembling a summary. “Aside from its immediate, stunning impact on the financial markets, the recent 25% drop in crude oil prices may have longer-term implications,” said PIMCO portfolio manager Greg Sharenow in a note. “We expect crude oil to remain under \$40 per barrel (bbl) for some time, with risks to fall materially lower should the production increases endure for multiple quarters. This will have repercussions for many industries, notably U.S. shale oil, and for consumers around the world.”

The marginal cost for lifting oil for the Saudis is very low relative to others, and its reserves are much higher as well.

As for marketplace war with the US, the war is with the bondholders and shareholders of shale companies. If the shale developer declares bankruptcy, the assets are still in the ground...so the developed resource will still be extracted.

So, the real target is those who are deciding to invest in additional shale development projects. The bankrupt ones will produce under new ownership and with new financing.

During the late 70's and early 80's there were many studies of commodity cartel economics. If you go to the FTC website and go the the Bureau of Economics you can find some of these papers. Here is a current OPEC related paper on the magnitude of misallocation: https://www.ftc.gov/system/files/documents/public_events/1208143/collardwexleraskerdeloecker.pdf

Shale oil and gas is the red queen's race - the wells play out after a year or two, requiring constant new drilling to simply keep up the same output. A pause of six months in new drilling means a predictable fall off in production later. Basically, it is not a matter of the resources in the ground, it is a matter of the resources (that is, the men, materials, and rigs) to keep drilling new wells.

The Saudis are fully aware of this, and thus have little incentive to end their price war in short order. One could make the argument (with some if not total justification) that the American fracking industry was more or less a pyramid scheme - and the Saudis may have decided it is time to tear it down, if only to put paid to the idea of Saudi America.

Thanks. So, the question is, after bankruptcy, and a purchase from the bankrupt estate (purchase price reflecting discounted cash flows) whether the the price will rise, and whether even at low prices, the marginal costs are below the market price. Also, as for resources to restart (men, materials, rigs) there is a market for them, and there are suppliers of those products or human resources.

What are the marginal costs and are they below the market price.

Who will be the company that rolls up the bankrupt companies? I'd invest in it as in the long run it is in neither Saudi or Russia's interest to have low prices. They just have to get used to the idea that there always will be a competitive fringe that will expand output if they raise their prices too high.

Basically, the domestic shale oil industry cannot make money when oil is below \$50 dollars a barrel, though that price does not lead to immediate bankruptcy.

The problem is that as long as the price is under that amount, no one will be buying or using any of that drilling capacity. And since Trump just singlehandedly removed about 600,000 barrels a day used by the airlines flying to Europe, the market may continue to fall even further.

Or not - who knows? But a year of oil under 50 dollars a barrel will be devastating for both the drilling industry, and the amount of oil that America produces. The Saudis would, as you recognize in terms of Saudi costs, be more than happy to sell America all the oil lost through that disruption at 45 dollars a barrel.

The Saudis are masters at this game, and have been for a half century. The Russians look like amateurs in comparison.

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Democratic Party insiders have banned moderate candidate Representative Gabbard from the next Democratic candidates' debate. The move is widely seen as an attempt, by corrupt insiders, to impose Joseph Biden, a decades long Washington insider, as presidential candidate.

Oh no! Private organization creates its own rules to prevent people with almost no chance of winning the nomination from being a spoiler though she also has almost no chance of being a spoiler either. They need some sort of threshold to limit the number of participants at debates and if she can’t even meet those minimum requirements when there are just two other candidates it’s a waste of time.

"They need some sort of threshold to limit the number of participants at debates"

Why? How dis they decide two (old men) was the magic number?

"and if she can’t even meet those minimum requirements when there are just two other candidates it’s a waste of time."

She actually won delegates. Let us be blunt: Washington insider Joseph Biden is trying to steal the nomination the way Clinton did in 2016 (the outcomes we all can remember, except Biden maybe).

Shh! They need Joe, a racist old creep, to win.

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As I watch the hysterical coverage on news programs, it’s becoming clear to me that the prophylactic measures are causing far more harms than the virus.

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Many St. Patrick's Day celebrations this weekend are still scheduled, even in large cities. Unless those get cancelled soon, I expect a large number of people to get exposed this weekend and we'll so those people start displaying symptoms in another week.

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Do you mean a still remarkably ineffective American testing system?

Do you mean the still seemingly non-existent thermal scanners at all ports of entry in the U.S.?

Do you mean effective quarantining, which the release of people still infected with covid-19 indicates a learning curve?

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Thanks Alex - we've developed a tool to provide a more user friendly data input interface for doing the math.

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I would think that in fact transmission is a bigger problem on a 100 person jet over a three hour flight, than in a 20,000 person stadium.

Doesn't intimacy and duration of contact, length of time before re-use, air circulation, etc. matter more than sheer numbers?

I mean, what they hell does it matter of one person is infected in a seat 100 yards away across a stadium, with thirty stairwells and fifteen batrooms? Is that really any worse than 300 people in a nightclub who share one front door, one bar, and one bathroom?

It's probably a function of number of contacts and duration/intensity of exposure.

Also, in Wuhan, Italy, SK etc, we're talking about "events" that last more than a day - so that infected people can develop symtoms and infect others over the course of a week. Chinese New Year, Mardi Gras, repeated attendance at the same church.

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Yeah, I come into extended personal contact with everyone at a dinner party of 10.

In a large venue, think of a beer vendor or ticket-taker as a promising vector.

Still at some point, number of contacts is going to overwhelm intensity of contact. Moving around in a crowd of 10,000 is just going to expose more people than moving around in a crowd of 1,000. An infected person can only infect a maximum of 10 people at that dinner party.

Perhaps. But perhaps not. And probably not on a one-for-one scaling.

In the end, one infected person in a 20,000 seat stadium still only infects those he comes within 6 feet of (so to speak). And they all go home before they can infect anyone else.

It is in fact entirely possible to get all 20 attendees at a dinner party well-and-fully exposed over a two hours of intimate interaction, particularly if you are the host and prepared the food, napped on the couch, etc..

On the other hand, one infected person attends a large stadium and only has fleeting or arms-length contact with a half dozen people tops. Maybe another handful for the ten seconds when they are squeezed through the gate. But yeah, if they grab the flush handle or mustard dispenser and no one washes their hands....

On the other hand, one infected person attends a large stadium and only has fleeting or arms-length contact with a half dozen people tops.

What he doesn't do any standing in line for the bathroom, entrance, concession stand? Or when leaving after the game ends? That part typically takes longer than 10 seconds. More like 10 minutes .... if that. Not that I've been to a concert or sporting event in a while, but I do recall standing in crushing crowds filing out slowly over 10-30 minutes.

Well, yeah, it all depends. Which is the point I am getting at: it's not a linear or homogenous curve, except for the most basic illustration. But if we seriously want to stop this thing, we shouldn't just cancel events > 2,000, we should ban dinner parties.

No, it doesn't depend. There's more risk of infecting people at large crowded events. Sorry.

Yes. absolutely. You are sorry.

I think it's hard to say . I have been to sports events where I haven't interacted much but it can happen. At dinner parties, there's also quite a lot of intimate contact. Also an infected person who doesn't feel so sick may not shed a lot of virus. Historically ( SARS/MERS) a lot of super spreaders were in hospitals, sick patients with a high virus load, shedding a lot of virus in a not so well ventilated/protected area.

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This is a decent framework, but making it work at a US scala is kind of crazy. The chances of an attendee being a carrier are easily affected by 3 orders of magnitude by state. This compounds with population density: The number of people one walks by in a day in Madrid has little to do with the number in, say, Hannibal Missouri. So sure, I'd be cancelling wedding receptions in Seattle, but the local high school in Hannibal is probably fine.

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Plus there seems to be no variable for how contagious a carrier may be at any given time. With many diseases, knowing such periods are considered reasonable when assessing transmission risk.

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Doesn't the formula assume an even geographic distribution of carriers?

So far Ohio has 4 confirmed cases. One nursing home in Washington had more fatalities than that.

Maybe the model assumes that people travel to national events.

Did you notice that colleges are distancing students (online classes) after the students come back from Spring break visiting other parts of the US.

Wonder why? Now you know.

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So yeah, probably a good idea to cancel march madness, and any concerts larger than 1000 people - especially if they are in Seattle or New York.

At a MINIMUM, have people with thermometers at the entrances and bar anyone with a fever from entering.

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By the time you are running a calculator like this, you'll find no-one is showing up anyhow. Restaurant cancellations are soaring, schools are closing, playgroups are empty.

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I suppose this could be termed The Wisdom of Small Crowds

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Good to see the basketball world taking a leadership position. Thank you, David Silver.

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I don't find this particularly useful. What's the probability of contracting COVID-19 given x infections among n people in area y?

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The majority of coronavirus infections may be spread by people who have recently caught the virus and have not yet begun to show symptoms, scientists have found.

An analysis of infections in Singapore and Tianjin in China revealed that two-thirds and three-quarters of people respectively appear to have caught it from others who were incubating the virus but still symptom-free.

The finding has dismayed infectious disease researchers as it means that isolating people once they start to feel ill will be far less effective at slowing the pandemic than had been hoped.

“This is one of the first things we were worried about when the outbreak began,” said Steven Riley, professor of infectious disease dynamics at Imperial College London, who was not involved in the work. “It was certainly unhoped for. This is one thing we really didn’t want to go this way.”

Researchers in Belgium and the Netherlands drew on data from outbreaks in Singapore and Tianjin to work out the “generation interval” for Covid-19. The generation interval is the time between one person getting infected and them infecting another. The figure is valuable for estimating the speed at which an outbreak will unfold.

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This presumes cases are randomly distributed. Which is just dumb.

If carriers are clumped, the lower risk of infection from events outside the cluster is largely offset by the much greater risk of infection from events in the cluster. And since carriers are almost by definition unknown (why are they going to events of any kind if anyone knows they are carriers?), the expectation of spreading infections at large events remains about the same. Think a little harder next time.

Do you suppose that a higher case rate in Italy affects rural Mississippi once everyone knows about the virus? Once travel bans are in place and events are cancelled.

Seems clearly not.

The probabilities may hold on average like you said, but its a useless exercise for any specific small - medium sized group or event.

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Trump made a mistake. He should have ignored Thiago, and banned Brazilians - "A Brazilian official who met President Trump and Vice President Pence on Saturday in Florida has tested positive for the novel coronavirus.

Fabio Wajngarten, communications secretary to Brazilian President Jair Bolsonaro, has been diagnosed with covid-19, Brazilian officials said — potentially bringing the illness into proximity with Trump."

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I think we should install in public bathrooms mirrors that have digital readouts of the temperature of the person looking in the mirror.

If no one has patented this idea, it is now public knowledge and you can't get a patent on it because it is obvious to any reader of this site.

Oh my, Bill, you're hot.

An obvious next feature to add to smart watches.

You're welcome, please connect me offline for how to send me royalty checks.

The check's in the mail.

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I found this post because I've had such a hard time recently with the following question: what's my current moral obligation in this crisis when it comes to very small scale interactions?

There are some obvious things: it's my obligation to wash my hands as frequently as possible. It's my obligation to avoid crowds of hundreds and thousands of people, to work from home if I can, etc etc, and these are things I'm doing.

But certain people in my group of peers have started saying that it's selfish to go out at all for interactions that aren't necessary; they say that things like going out to restaurants or having a party with 12 people over at my house are selfish, and I don't particularly like being called selfish. I understand their argument that going out at all places yourself and others at increased risk, and to do so is selfish. My view is that, from posts like this and others, the difference in likelihood of spreading the disease between staying completely inside my house and interacting in small groups is so minor, and it only makes a difference whether I'm going out in large groups, which I'm not. Perhaps I'm interpreting the data unreasonably to my own benefit, though.

This isn't a question for people who, for example, live paycheck-to-paycheck and don't have the choice to stay at home, but have to work in order to survive right now. This is a question for extremely privileged people like me who are showing no symptoms at all, are not around people with symptoms (but are in a city with cases), and don't think it's a moral obligation right now to stop any sort of "unnecessary" small-scale social interaction.

Am I wrong?

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I've implemented the risk model in this post into a google sheets spreadsheet. you can view or copy it from the link below.

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It's somewhat misleading, it's really hard to know how many active circulating cases there are. Once you know you are a carrier and you decide to self quarantine you are out of the equation. Would be interesting to see an estimated number of active circulating cases that are not limiting their contact with others.

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Does anyone know a source -- journal article, monograph, textbook, etc. -- for the formula used in this post?

I'm completely unfamiliar with contagian network theory. Is that discipline the source of the formula?

Since I don't even know where to start, any help would be appreciated.

Thanks.

This is the formula I'm talking about:

probability of exposure=1-1(1-c/p)^g

where p is the population size, c is the number of carriers, g is the group size

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